


\magnification = 2000 
\input amstex
\documentstyle{amsppt}
\input OurATOMacros
\input OurPlainGraphicsMacros


\vglue 10pt

\Title Part I:  Platonic Solids.
\cl{\bf Relations among them, Simple Truncations} 
\cl{See Part II below: Archimedean Solids} 
\LF
There are five (and only five) Platonic solids. Three are easy to imagine---the Cube, 
Octahedron and Tetrahedron---while the remaining two are more difficult: the Icosahedron
and Dodecahedron. The earliest known models date from the Stone Age.

{\sc What to do in 3D-XplorMath?} \lf
First, the program shows how the other four Platonic solids are obtained from
the Cube: Select first one of the other polyhedra, then in the Action Menu:\lf
\phantom{.\hskip1.5cm} {\tt Show Relation with Cube}.
\lf
For the Octahedron one sees that its six vertices are the midpoints of the faces
of a cube; the Octahedron faces are equilateral triangles. The Tetrahedron sits
in the cube so that its four vertices are vertices of the Cube, and the six Tetrahedron
edges are face diagonals of the Cube.  The Icosahedron can be placed inside
a Cube so that its twelve vertices lie on the six faces of the Cube: see the default
morph in 3DXM, preferably when {\tt Show Relation with Cube} is chosen.  The
Dodecahedron can be placed around a Cube so that its twelve pentagon faces
rest on the twelve edges of the cube. See again the default morph going from
the Rhombic Dodecahedron via the Platonic Dodecahedron to a cube with 
subdivided faces.
\LF
Second, by cutting off appropriately the vertices or the edges of a Platonic solid
one obtains the simpler {\bf Archime\-dean solids}. These are polyhedra whose faces 
are (several kinds of) regular polygons, whose edges all have the same length and
whose vertices all look the same. Select in the Action Menu any of the three
{\tt Truncations} and then do the associated default Morph. For example, the
{\tt  Edge Truncations} morph from one Platonic solid to another one (which is
called its dual).
\LF
Third, since our intuition handles two dimensions better than three, it is
interesting to project the Platonic solids from their midpoints onto a circumscribed
sphere. View in Wire Frame and select in the Action Menu:\ \ 
{\tt Show Central Projection to Sphere}.\lf  These two-dimensional 
spherical views of Platonic solids come very close to explaining why 
there are no other such beautiful polyhedra.
\LF
Fourth, the Icosahedron and the Dodecahedron have very beautiful 
Stellations, polyhedra that fascinated Kepler. Select (Action Menu): {\tt Create Stellated}.
Note that all the mentioned views have their own default morphs. In addition,
one can select in the Polyhedra Menu {\tt Kepler's Great Dodecahedron}. Kepler
viewed its faces as Pentagon Stars. In 3DXM it is drawn as a (negative) stellation
of the Icosahedron.
\LF
Fifth, for the Cube and the Icosahedron there are two special entries in the 
Action Menu when viewing these solids in Wire Frame.  For the Cube select
{\tt Show Intersection With Plane}, preferably in one of the stereo modes. The
plane is represented by random dots and the dots inside the Cube are deleted;
the Cube can be rotated and moved forward and backward, always showing
its polygonal intersection with the plane. For the Icosahedron select in the
Action Menu {\tt Add Borromean Link}, preferably in one of the stereo modes.
Note how the boundaries of the emphasized rectangles are intertwined or
linked. The edge lengths of each rectangle are equal to the lengths of an 
edge and a diagonal of a regular pentagon, thus showing the relation of the
Icosahedron inside the Cube with the Golden Ratio. Also, the default morph
of this image is worth viewing.
\LF
Finally, stone objects with Platonic Symmetry were found, mainly in Scotland. 
They were dated 2500 B.C. They are carved from bigger stones, but they look
as if they were conceived as collections of balls. Therefore we have added the
Action Menu entry: {\tt Show  As Stone Balls}. In Patch Display the balls are fine
triangulations of the Bucky Ball, in Wire Frame the balls are shown with random
dots. One can view other sphere triangulations after one has selected (in Patch
Display) {\tt Create Subdivided}: another entry appears: {\tt Triangulate Further}.
\LF
\bigskip
\goodbreak
\cl{\bf Part II: Archimedean Solids} 

\noindent
Here is the definition again: All faces are regular polygons (of up to three different
kinds). All edges have the same length. All vertices (with their outgoing edges)
are congruent. In addition to the five Platonic solids there are twelve of them. \lf
We have already seen the simplest ones: Truncate the vertices of a Platonic
solid; there are two possibilities, if  some portion of the edges remain this is
called {\tt Standard Truncation}, and if the truncation cuts go through the midpoints
of the edges we have a {\tt Midpoint Truncation}. One can also {\tt truncate the edges};
this deformation leads to the same Archimedean solid if one starts from the 
Octahedron or the Cube, and also if one starts from the Icosahedron or the 
Dode\-ca\-hedron. Two more are obtained if one truncates the edges and the
vertices; the deformation is easier to observe if one uses the standard
truncation on either the Cubeoctahedron or the Icosidodecahedron. In fact,
not quite the standard truncation, because that would make rectangles instead
of squares from the truncated vertices. \lf
Finally there are the {\tt Snub Polyhedra}. We could not find what 'snub' means
in this context. We describe the construction and call it 'to snub'. Each face of a
Platonic solid is scaled down from its midpoint and also rotated around the midpoint.
The 'snubbed' po\-lyhedron is the convex hull of these deformed faces. This
2-parameter deformation can be adjusted to give a 1-parameter family of
polyhedra whose faces are either regular polygons or isosceles triangles. In
each family is an Archimedean solid. A snubbed Tetrahedron is an Icosahedron,
snubbed Cube and Octahedron give the same Archimedean solid and also
snub-bed Dodecahedron and Icosahedron agree.
\LF
One can probably understand all these truncations better if one selects in the
Action Menu \lf
\phantom{1.}{\tt Snub Or Truncate Polyhedron In Polyhedron}. \lf
This will add the original polyhedron (as wire frame) to the truncation. \lf
Note also that each selection in the Action Menu will cause the default
deformation, {\tt Morph} in the Animation Menu, to be adjusted to the Action Menu
selection.



\noindent
H.K.



\bye

